Abstract
Abstract
In this paper, we study the linear stability of the elliptic rhombus solutions, which are the Keplerian homographic solution with the rhombus central configurations in the classical planar four-body problems. Using ω-Maslov index theory and trace formula, we prove the linear instability of elliptic rhombus solutions if the shape parameter u and the eccentricity of the elliptic orbit e satisfy
(
u
,
e
)
∈
(
1
/
3
,
u
2
)
×
0
,
f
^
(
27
4
)
−
1
/
2
∪
(
u
2
,
1
/
u
2
)
×
0
,
1
∪
(
1
/
u
2
,
3
)
×
0
,
f
^
(
27
4
)
−
1
/
2
where u
2 ≈ 0.6633 and
f
^
(
27
4
)
−
1
/
2
≈
0.4454
. Motivated on numerical results of the linear stability to the elliptic Lagrangian solutions in Martínez et al (2006 J. Differ. Equ.
226 619–651), we further analytically prove the linear instability of elliptic rhombus solutions for
(
u
,
e
)
∈
(
1
/
3
,
3
)
×
0
,
1
.
Funder
Sino-German (CSC-DAAD) Postdoc Scholarship Program
National Natural Science Foundation of China
Innovation Program of Shanghai Municipal Education Commission
Science and Technology Innovation Action Program of STCSM
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics